Demonstrating the saturation of stimulated Brillouin scattering by ion acoustic decay using fully kinetic simulations

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1 Demonstrating the saturation of stimulated Brillouin scattering by ion acoustic decay using fully kinetic simulations T. Chapman,, a) B. J. Winjum, S. Brunner, R. L. Berger, and J. W. Banks 4 ) Lawrence Livermore National Laboratory, P.O. Box 88, Livermore, CA 9455, USA ) Department of Electrical Engineering, University of California Los Angeles, Los Angeles, California 995, USA ) Centre de Recherches en Physique des Plasmas, Association EURATOM-Confédération Suisse, Ecole Polytechnique Fédérale de Lausanne, CRPP-PPB, CH-5 Lausanne, Switzerland 4) Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Amos Eaton Hall, Troy, NY 8, USA (Dated: April 5) The saturation of stimulated Brillouin scattering (SBS) by the decay to turbulence of the ion acoustic wave (IAW) that participates in the three-wave SBS interaction is demonstrated using a quasi-noiseless one-dimensional numerical solution to the Vlasov-Maxwell system of equations. This simulation technique permits careful examination of the decay process and its role in the complex evolution of SBS. The IAW decay process is shown to be an effective SBS saturation mechanism; in our example, the instantaneous plasma reflectivity saturates at % and drops to % as a direct consequence of IAW decay. A contrasting example where the reflectivity is controlled by dephasing due to the nonlinear frequency of the IAW is also discussed. I. INTRODUCTION Stimulated Brillouin scattering (SBS) persists as a problematic source of backward-scattered light in indirect-drive inertial confinement fusion (ICF) experiments. In ICF experiments, plasma near the wall of hohlraum targets and in the ablated outer layer of the fuel capsule is observed to produce levels of SBS that are typically energetically weak compared to the total laser energy delivered to the hohlraum. However, when arriving in bursts, this backscattered energy is sufficient to damage the sensitive and expensive optics required at facilities such as the National Ignition Facility (NIF) and remains a central consideration in experimental design. In the SBS process, laser light scatters from and drives ion acoustic waves (IAWs). IAWs are weakly dispersive, permitting coupling between modes with wave numbers that differ greatly and making IAWs susceptible to decay. IAW decay has been observed directly in dedicated Thomson scattering experiments,4 and correlated with the saturation of SBS. Numerical studies have identified IAW decay during SBS saturation 5 9, while other work has examined IAW decay in isolation and demonstrated the role of electron kinetic effects in IAW nonlinearity,,,4. In Ref., the efficient modemode coupling of IAWs was found in simulations of both free and continuously driven IAWs to result in a decay process that occurred readily across much of the parameter space of relevance to ICF experiments. Study of IAW decay has been motivated in part by the potential of this decay process to saturate the SBS instability. The decay of the IAW driven during SBS a) chapman9@llnl.gov would naturally act to suppress further scattering of the laser light, thereby limiting the reflectivity (defined as the scattered light intensity divided by the applied laser intensity) of the plasma. The numerical results of prior work 5,8,9,, suggest that IAW decay can indeed lead to a highly turbulent nonlinear state. The decay process does not conserve energy stored in the field of the IAWs, resulting in particle heating and a nonlinear IAW fluctuation level well below that predicted by simple fluid models. In the current work, we investigate the role of IAW decay in SBS saturation using the kinetic code sapristi 4,5, which solves here the collisionless DV (one spatial and one velocity dimension) Vlasov-Maxwell system of equations using a continuum method. Our simulations include the kinetic behavior of electron and ion species, both of which play important roles in the nonlinearity of IAWs. The presented simulations are designed to distinguish the saturation of SBS via IAW decay from other saturation mechanisms, such as depletion of the laser light (or pump ) due to scattering and dephasing due to a nonlinear frequency shift of the IAW. Our simulations demonstrate the effective saturation of SBS by the decay of the primary IAW to subharmonic modes (i.e., modes that do not have wave numbers that are integer multiples of the fundamental). In addition to decay-generated subharmonics, significant IAW harmonic generation 4 is also observed, a process which has been found to be important in reduced models of SBS 6,7 and may influence IAW decay rates 9,,. In our simulations, we seed SBS but not the stimulated Raman scattering (SRS) process. Due to the quasinoiseless simulation technique employed, we do not observe SRS. This approach allows the study of SBS in isolation from the SRS process.

2 In the following, plasmas will be categorized by the parameter α = ZT e /T i. This parameter determines the strength of the ion Landau damping of IAWs in the linear regime (i.e., before significant particle trapping has occurred, which acts to suppress Landau damping). In ICF experiments, laser light interacts with both low-z hohlraum gas fill (He) and capsule ablator (CH, Be, C) materials and high-z hohlraum wall materials (Au, U). Laser pulse lengths are sufficient for electron-ion temperatures to nearly equilibrate until peak power is reached. As a result, plasma conditions vary across 4 α. IAW decay was found previously 5, to behave similarly in both single and multi-ion species plasmas; here, we study a single ion species plasma (He) in regimes characterized by values of α that differ greatly. Our key findings are summarized as follows: When the decay IAW modes of the fundamental IAW mode that is resonant with the SBS interaction become comparable in amplitude to the fundamental mode, the fundamental mode collapses and a highly turbulent state that is only weakly reflective ensues. The decay process is qualitatively described by a fluid-like decay of the IAW to two daughter IAW modes, known as two-ion wave decay, and was observed to be weakened by increasing α in a D system. At high α (we show the case α = 5) where ion kinetic effects are negligible, ion wave decay is suppressed. In this case, the evolution of the reflectivity is dominated by the dephasing of the driven IAW from the ponderomotive drive of the light waves 7,8. This dephasing is due to the trapping-induced nonlinear shift of the IAW frequency away from its linear value, dominated at high α by electron trapping 4. The layout is as follows: The physical motivation behind our choice of plasma parameters is given in Sec. II. In Sec. III A, simulations are presented showing SBS saturation via IAW decay and the resulting particle heating in Sec. III A. In Sec. III B, simulations are presented showing SBS saturation by the nonlinearity of the IAW frequency. Finally, in Sec. IV, we discuss our results in the context of relevant prior work. II. MOTIVATION OF PARAMETERS Our goal is to demonstrate the role of ion wave decay as a saturation mechanism of SBS independent from other potential saturation mechanisms. To this end, two distinct cases are addressed in this work. The parameters for these cases, labeled set S and S, are summarized in Table I. Significant prior simulation studies have been accomplished in which IAW decay has been studied explicitly or simply observed to occur; prior work is discussed in the context of the new results presented here in Sec. IV. An IAW driven during SBS must satisfy matching conditions in frequency and wave number between itself (subscript s), the laser light (subscript ), and scattered light (subscript ): ω = ω + ω s and k = k + k s, where TABLE I. Summary of analytically-determined parameters Set ZT e/t i v φ /vti ν s/ω L s δω s I x= (W/cm ) S S For both sets, Z =, T e = kev, m i = 4u ( 4 He), n e/n c =., k sλ De =.75, L = λ λ s, λ = 5 nm, ω = ω ωs L, and I x=l = I x= / 6. Obtained from solving the KDR. Here, v φ = ωs L /k s. ωs NL = ωs L + δω s, where δω s/ωs L = δω s eφ/te. See Refs. 4,, and 4. here all quantities are real scalars and defined as positive but for k < (i.e., the SBS light is scattered backwards). We consider a plasma composed of species j with density n j, charge Z j e, mass m j, and temperature T j, where e is the magnitude of the electron charge (note Z e and Z i Z throughout). Subscripts of e and i indicate electron and ion quantities, respectively. The species Debye length is given by λ Dj = v tj /ω pj, where ωpj = n j Zj e /(m j ε ) is the species plasma frequency, and v tj = T j /m j the thermal velocity. ε is the permittivity of free space. The complex IAW frequency Ω may be obtained by solving the kinetic dispersion relation (KDR) numerically. For He with Maxwellian species distributions, good analytic approximations are possible: The real mode frequency ω = Re(Ω) is, [ ω c i k ( + k λ De ) + ] /, () α where c i = ZT e /m i is the cold ion sound speed. The Landau damping may be decomposed into electron and ion contributions ν Im(Ω) ν e + ν i, where, Zme ν e ω ν ν i ω να/ exp, () m i ( α ), () and ν = [(π/8)/(+k λ De ) ] /. Note that the quantities k, ω, and ν refer to an unspecified IAW mode, while the subscript s denotes the specific IAW mode that is resonant in the linear -wave SBS interaction. The laser intensity threshold for SBS is proportional to ν s /ω s (see, e.g., Ref. 5) and is highly sensitive to α [see Eq. ()]. Accordingly, the expected linear SBS response in cases S and S may differ substantially for a given laser intensity. In our simulations, the light waves are effectively undamped (apart from at the boundaries) so there is no damping threshold for convective growth. Case S lies below the threshold for absolute instability. Case S is slightly above this threshold, but the absolute instability growth rate is small compared to the convective instability growth rate. The convective gain is given by G conv = Γ L/(cν s ), where Γ is the SBS growth

3 rate 5. For S, G conv = 6 while for S, G conv. Discussed subsequently, the applicability of such linear theory is limited due to the early onset of nonlinearity in our simulations. In this work, Φ = Φ(x, t) is the actual IAW potential, while φ = φ(x, t) is the amplitude of the envelope of Φ and typically varies slowly compared to the IAW period. It is often convenient to decompose Φ into Fourier components, Φ = l (/)φ l exp[i(k l x ω l t)] + c.c. (4) While the IAW driven during SBS remains approximately linear, one has φ φ s. The traveling potential interacts with resonant particles, i.e. those with velocities close to v φ = ω s /k s. In addition to Landau damping, resonant particle (kinetic) effects include a nonlinear (amplitudedependent) frequency shift of the IAW that is a consequence of particles becoming trapped in the wave potential. We differentiate between the linear IAW frequency ω s (φ ) = ωs L and nonlinear frequency ωs NL, for given wave number k s, by writing, ω NL = ω L + δω. (5) In a D system, particle trapping will also reduce and may eventually eliminate Landau damping, and is therefore important to the behavior of SBS near threshold. In the simulations presented in the following, species distributions are described by the continuous distribution functions f j = f j (t, x, v), where t, x, and v are scalars indicating time, spatial position and velocity, respectively (see Appendix for details). Trapping results initially in a flattening of f j λs in the resonant region of velocity space. This flattening is centered about v φ with a characteristic half-width, v tr,j, that may be estimated analytically to good accuracy for wave amplitudes that remain in the perturbative limit 4,,4. For electrons under the conditions discussed in this work, resonance occurs in the center of the distribution (v φ /v te ), and v tr,e /v te = eφ/t e ; for ions, resonance occurs in the tail of the distribution (here, v φ /v ti > ), and v tr,i /v ti = αeφ/t e. Note that v tr,j is a half -width, and trapping occurs within the regions bound by v φ ± v tr,j. An analytic estimate of δω is possible when the deviation of the distribution from Maxwellian remains small 4,,4. When this approximation is valid and the IAW is monochromatic, one has δω φ. Similar to the Landau damping, both electron and ion species contribute to δω, although these contributions are of opposite sign. Both the sign and magnitude of δω are therefore strongly dependent upon α. An analytic calculation of the frequency shift is possible for both adiabatic 4,4 and sudden 4,,4 excitation limits. Here, an adiabatic limit is appropriate for the electrons. The sudden limit is appropriate for the ions in S, while for S, the ion contribution is negligible 4. Total δω s is given in Table I. The other nonlinear property of IAWs that is important to this study is the decay of the k s mode. A simple resonant IAW decay model of a mother to two daughter waves is derived in Ref., which gives, γ a = γ b = c i eφ s ka k b T e, (6) where γ l is the exponential growth rate of the k l mode, and the subscripts a and b label the decay (daughter) modes that fulfill k s = k a + k b (and ω s = ω a + ω b ). This equation is obtained by a simple -wave resonant IAW decay model and includes no kinetic effects, and does not capture the dependence of γ l upon α found in Ref. ; however, it provides a useful benchmark for decay rates measured in simulations. In fluid treatments of IAWs, the quantity k s λ De determines the relative amplitudes of the harmonics of the fundamental IAW,4, although such treatments underestimate significantly the strength of harmonic generation present in fully kinetic simulations,4. In S and S, k s λ De =.75. The decay mechanism of IAWs was found previously to be weakly if at all sensitive to k s λ De. The minimum set of effective parameters in a collisionless single-ion species plasma is simply α = ZT e /T i and Zm e /m i. The cases S and S are distinguished primarily by the value of α used in each: α = 7 and α = 5 (in practice, we vary T i ). In S, the linear Landau damping is significant (of order 5% of the IAW frequency), the IAW decay rate is close to a maximum, and the nonlinear frequency shift is negative and of relatively small magnitude 4. In case S, the linear Landau damping is negligible (less than % of the IAW frequency), the IAW decay rate is strongly suppressed, while the nonlinear frequency shift is positive and of relatively large magnitude 4. In the simulations, the initial electron temperature and density are T e = kev and n e /n c =., respectively, typical of plasma with significant levels of SBS at the NIF. We do not impose gradients, external fields, or a plasma flow. In order to study the role of IAW nonlinearity (rather than pump depletion) in SBS saturation, the differing values of α in S and S necessitate differing laser intensities, chosen so as to produce similar IAW amplitudes and reflectivities in the two cases. The laser intensity is set to.5 5 and 7 4 W/cm in sets S and S, respectively, with vacuum wave length λ = 5 nm. The plasma length L is chosen to be L = λ λ s 5λ De, where λ s = π/k s. This length is short enough that the pump does not deplete strongly when undergoing SBS, but large enough that excited IAWs can undergo significant decay before advecting to the boundaries of the system. III. SIMULATION RESULTS The simulation setup is summarized in Fig., while a fuller discussion of the simulation technique is given in the Appendix. In both cases, the progression in simulations is as follows: A weak monochromatic electro-

4 4 magnetic seed of intensity I x=l, frequency ω, and wave number k exists throughout the plasma, introduced at the x = L boundary and propagating in the direction of decreasing x. A laser is introduced at ω pi t and x =, propagating in the direction of increasing x with intensity I x=, frequency ω, and wave number k. The light waves drive an IAW via SBS at (ω, k) (ω s, k s ), modifying f j in the resonant region of velocity and reducing the Landau damping. The seed is amplified via backwards scattering of the laser light across L and consequently I is typically largest near the x = boundary. Using the parameters listed in Table I, the simulation duration is approximately 4/ω pi.4 5 /ω pe 4 ps. The IAW amplitude is largest slightly further from the boundary due to the Krook damping layer (see Appendix) and wave advection. The maximum IAW amplitude grows in tandem with the reflectivity of the plasma, R I x= /I x=, until saturation due to nonlinear processes. In figures and in the text, normalizations of physical quantities are given explicitly. The exception is the electric field E (either longitudinal or transverse), which is normalized as Ẽ = eλ DeE/T e, where E = E (x, t) and E = E (x, t) are the transverse electric fields of the pump and scattered light, respectively. E x = E x (x, t) is the longitudinal electrostatic field of the plasma waves; E x (k) = E x (k, t) denotes the longitudinal spatial Fourier transformation component of E x with wave number k. A. Set S: ZT e/t i = 7, larger ν s, and smaller δω s. Reflectivity and subharmonic growth This case clearly exhibits the saturation of SBS via IAW decay, with no significant complicating factors. The full plasma parameters for the case discussed in this Section are given in Table I, listed under set S. In Fig., three snapshots in time of the longitudinal electrostatic field (E x, the IAW) and envelopes of the transverse electromagnetic field intensities (I and I ) are shown. In Fig. (a), the fields are presented just before the onset of saturation. Up to this point, the dynamics of the IAWs are essentially linear. Between the time snapshots in Figs. (a) and (b), the k s IAW mode rapidly decays to its subharmonics, leading to an increasingly Ion acoustic waves x= I, (ω, k ) in plasma x=l I, (ω, k ) x = x = L = λ 5λ De FIG.. Basic simulation geometry showing the pump laser (subscript ) and EM seed wave (subscript ). Dashed lines indicate the furthest extent of the plasma boundary damping layers (at scale). The EM antennae are within these boundary layers. turbulent plasma. This phenomenon confirms the findings of Ref. for free-wave and fixed-driver simulations of IAWs. The moment at which the IAW amplitude (and plasma reflectivity) is lowest is shown in Fig. (c). It is apparent in Figs. (a-c) that SBS amplification occurs only when E x is essentially monochromatic. The pump [shown also in Figs. (a-c)] is only weakly depleted throughout. Turbulent regions of plasma are minimally if at all reflective. The plasma reflectivity, R, and evolution of E x in k- space are shown in Fig. (a). R peaks at.5 then falls to 4. The saturation and subsequent crash of R coincide with the onset of IAW turbulence 5,8,9. After crashing, R begins to grow again; we attribute this recurrence simply to the advection of IAWs and resonant particles out of the simulation box, resetting the system to near its initial quiescent state. The phenomenon of recurrence in SBS has been observed under similar simulation conditions using a PIC code 8, as well as in the study of SRS saturation using Vlasov 6 and PIC 7 codes. The time taken for IAWs and resonant particles to reach the boundary of the system is at most τ L/c i 5/ω pi, in rough agreement with the recurrence period τ rec observed in Fig.. The particle distributions may however retain remnants of their perturbations, even in the absence of any further driving of IAWs; the bulk with v tj is also somewhat modified during SBS, and these modifications may propagate slower than c i. In Fig. (b), the frequency ωs NL of the IAW k s mode that is linearly resonant with the pump and seed and the frequency ω max of the largest amplitude IAW mode are shown as deviations from the linear resonant mode frequency, ωs L. In simulations, the instantaneous frequency of a mode with a given wave number is calculated by taking the time derivative of the mode phase θ, where θ is the angle of the complex phasor of Ẽx(k). Early in time, one sees ω max = ωs NL ωs L (i.e., k max = k s ), as expected for a regime where the plasma waves remain only weakly nonlinear. Later in time, ω max and ωs NL begin to diverge. Shown also is the value of δω given by theory, specified in Table I and using φ = Φ L. Good agreement between theory and simulations is not expected, since φ varies significantly in space and furthermore the theory assumes a weak and monotonically increasing perturbation of the distributions from Maxwellian. However, up to the first peak in R, the measured nonlinear frequency shift is negative and qualitatively consistent with theory. At the first peak in R, δω is of the order of only.5%. As a consequence, we do not observe the characteristic beat pattern of a driven nonlinear oscillator present in the case S (discussed later). A more detailed analysis of the impact of the dephasing of the IAW from the ponderomotive force of the light waves is given in Sec. III B. The mode frequency is ill-defined when the plasma is highly turbulent, and this period in time is omitted from Fig. (b). In order to measure the linear decay mode growth rate,

5 5 I (a) Ex (b) ωpi tc = -. 5 (c) ~ Ex ωpi tb = τrec - ta tb tc ω/ωls (%) R, lin. scale ωpi t Theory ωnl s ωmax (b). 9 ~ Ex.5.8 R, log. scale.5 (a) 5 FIG.. (Color online) Snapshots in time of the longitudinal electrostatic field (IAW, left linear scale) and transverse electromagnetic fields (laser and scattered light, right logarithmic scale) using parameter set S. The transverse fields are shown as envelope intensities normalized to the input laser intensity, I. Turbulent regions in Ex correspond to greatly diminished local growth of I via SBS. Grey boxes indicate sampled regions used in Figs. 7 and 8. τfit.5 k/ks - [x - υφ (t - t)]/ λde -7 ~ Ex(k) x/λde Reflectivity, R x= I I /I ωpi ta = FIG.. (Color online) (a) Using set S, the reflectivity of the entire plasma versus time (right vertical axes, linear and logarithmic scale) and the changing composition of Fourier k modes in the longitudinal field of the IAW (left vertical axis and top color bar). The saturation and fall in reflectivity coincide with the onset of IAW turbulence, while the recurrence period τrec L/ci is determined by the time taken for IAWs to cross the plasma. The circled points ta,b,c correspond to the times shown in Fig. (a-c). (b) The deviation from the linear frequency of the resonant mode during SBS, ωsl, of i) ωsn L according to theory given in Table I (model si), ii) ωsn L from simulation, and iii) the largest amplitude IAW mode, ωmax, from simulation. it is necessary to select spatial and temporal windows that are i) small enough in x and t such that the spatio- 5 t..4 ωpi t FIG. 4. (Color online) The longitudinal electric field Ex in the wave frame moving at vφ.9ci, sampled across x t=t /λde = [5, ] where x = x vφ (t t ), ωpi t = 9, and vφ is a measured quantity. Period doubling due to the growth of the k = ks / mode (half harmonic) is apparent at ωpi t., just before the onset of turbulence. Note that there are no backward-propagating modes apparent in the laboratory (stationary) frame. temporal variations of φs are small, and ii) large enough such that the resolution in k-space is adequate and shorttimescale subharmonic mode amplitude oscillations do not distort the measurement. Ex in the wave frame for one such choice of window is shown in Fig. 4. At the onset of turbulence near ωpi t., period-doubling due to growth of the half-harmonic of the ks mode is apparent before the IAW collapses totally. Early in time, the IAW spectrum shown in Fig. (a) is dominated by the SBSdriven ks mode and its harmonics at kl = nks, where n =,,.... The subharmonic growth rate, γl, of the

6 6 E ~ x (k) γ l /ω s L k = k s k = k s /.9 tω.. pi τ fit Growth Decay (a) Modes driven resonantly via SBS Vlasov simulation Exponential fit Vlasov simulation -wave fluid theory (b) k l /k s FIG. 5. (Color online) (a) Mode amplitudes versus time for the SBS-driven k s mode and k = k s/ decay mode, obtained using set S. Exponential fits are made across τ fit using the moving window defined in Fig. 4. (b) Growth rates γ l measured using the fitting method shown in the upper figure. Mode growth is fastest near the half-harmonics of the k s mode, k l = (n /)k s. The -wave fluid theory is given by Eq. (6) using eφ/t e τfit =.. mode k l nk s may be extracted by fitting a linear slope to the logarithm of φ l (or E x (k l ) ) in the wave frame. Figure 5(a) shows this fitting process for the k s mode and its half-harmonic. Figure 5(b) shows γ l versus k l. The regression coefficients of the fits are of order.8, indicating that indeed φ l exp(γ l t) for k l nk s across the given time window. γ l displays the characteristic features of the two-ion decay process, namely a growth rate that as a function of k l takes the form of an inverted parabola, is maximal at and symmetric about k l = (n /)k s, and is periodic in k s. The blue dashed line is given by Eq. (6). Growth rates and scalings with both φ s and α were obtained for the decay process in Ref.. Despite the difficulties inherent to the measurement of γ l performed here (in particular, the non-uniform time-varying nature of φ s and corresponding irregular ponderomotive drive strength from SBS), the values of γ l measured during SBS are in agreement with Ref., where γ l was found to exceed the predictions of -wave fluid theory by a factor of for α = 7 (here, this factor is ). Note that the IAW subharmonic growth rate is significant, here taking a maximum value of the same order as the linear Landau damping rate (see Table I). The question of precisely why IAW decay rates often exceed those of a three wave fluid-like model in which the decay is assumed to be exactly resonant (as in Ref. ) has not been addressed satisfactorily by previous work and remains open. In Ref. it was found in the framework U/n e T e U ES K e U se U EM K i U si ω pi t FIG. 6. (Color online) For S, plot of the change in energy density U attributed to the longitudinal electrostatic field ( U ES, red line), the transverse electromagnetic field ( U EM, blue line), the electron kinetic energy ( K e, green line), and ion kinetic energy ( K i, black line). The jump in U EM at ω pit is due to the turning on of the laser. The green and black dashed lines are the electron and ion sloshing energies U se and U si, respectively, defined in the text after Eq. (7). of a fluid-like (Boltzmann) electron response that the decay of the second harmonic of a fundamental IAW mode could be dominant over the decay of the first, characterized by a subharmonic growth rate that scales with φ rather than φ (here, the subscript indicates the harmonic of the fundamental mode with wave number k ; e.g., the n th harmonic has wave number k n and amplitude φ n ), but clear support for this mechanism was not found in Refs. 9 or. In Refs. 4 and, it was found that nonlinear electron kinetic effects play an important (perhaps dominant) role in determining the amplitude scaling of φ n> with φ, suggesting strongly that subharmonic decay rates should also be sensitive to kinetic electron physics. The weak rate at which φ n / φ decreases with n, where all φ n harmonics may act as pumps for subharmonic modes with wave numbers k l n, combined with clear evidence of nonlinear electron and ion kinetic effects, presents an unsolved and substantial challenge for theoretical studies.. Particle heating In Fig. 6, the changes in electrostatic (E ES ), electromagnetic (E EM ), electron kinetic (K e ) and ion kinetic (K i ) energy are shown for S. In the units of Fig. 6, the initial electron and ion kinetic energies are K e /(n e T e ) =.5 and K i /(n e T e ) =.5/α =.74, respectively, where n e is the initial unperturbed electron number density. We are particularly interested here in particle heating via resonant IAWs. However, included in the K e,i is the sloshing of the distributions in response to E x, a process that is reversible in collisionless plasmas and does not contribute to heating 8. In order to determine the extent (if any) of the particle

7 7 heating, it is necessary to separate the heating from the sloshing energy. We define for the energy of a spectrum of IAWs in a Maxwellian distribution,8, U T n e T e = kl λ De l eφ l T e ω (ωɛ L), (7) ω=ωl where ɛ L = + χ e + χ i and ɛ L (Ω, k) =. It is convenient to use approximate forms of the susceptibilities χ j, χ e k λ, De ( ) (8) χ i ω pi ω + k λ ωpi De ω α. (9) Using these expressions to expand the right hand side of Eq. (7) analytically, one finds, ( ) ω (ωɛ L) + k λ + ω pi D ω + 9k λ ωpi De ω. α () After substitution of Eq. () into Eq. (7), one finds, where, U ES n e T e = U se n e T e = U si n e T e = U T = U ES + U se + U si, () kl λ De l l T e eφ l T e, () eφ l, () ( β l + 9 ) eφ l α β l, (4) l for which U sj is the sloshing energy of species j and β l = (k l λ De ω pi /ω l ). The heating of species j should be given approximately by K j U sj, where K j K j (t) K j (t = ) and K j = K j (t) = (/L) Γ dx dv (m j/)v f j is calculated directly in the simulations. U se and U si are calculated using the analytic expressions given by Eqs. (), (), and (4), with φ l taken from simulations. For a monochromatic IAW, one finds U se U si U ES /(k λ De ). However, as the IAW spectrum becomes turbulent, such a relation does not hold, although in general one has U sj U ES. If instead of the analytic expressions for U se and U si one uses the value of ɛ L given by solving the KDR to obtain ɛ L numerically, one finds for a monochromatic IAW with kλ De =.75 and α = 7 a change in the value of U se of.% and a change in the value of U si of 6%; i.e., the analytic expressions are adequate for our purposes. As resonant particle interactions become more significant, the distributions diverge increasingly from Maxwellian, decreasing the validity of Eqs. () and (4); this is discussed shortly. T e In Fig. 6 the electron and ion sloshing energies U se and U si are plotted as green and black dashed lines, respectively. In this case, U si diverges from K i almost from the outset, while U se K e until the onset of turbulence. After the onset of turbulence, it is clear that the change in kinetic energy in the system is dominated not by sloshing, but by what we refer to here as heating. This assertion may additionally be checked for the ions by verifying that the change in kinetic energy occurs primarily in the resonant region rather than in the bulk of the distribution, i.e. K res K i, where, K res dx L m i dv v f i, (5) x v>v and v v φ v tr,i. The relation K res K i is satisfied to within approximately 5% throughout the simulation. Such a check is not possible for the electrons, since the resonant region encompasses the bulk. K i increases from its initial value by 5% and K e,i continue to grow long after the reflectivity saturates. It is important to note that the increase in K e,i that occurs after the peak in U ES at ω pi t. in Fig. 6 can not be attributed to a simple conversion of U ES to K e,i via the decay of effectively undriven IAWs, since K e,i U ES ; IAWs are continuously driven despite the weak reflectivity during the turbulent phase, and their electrostatic energy is converted to kinetic energy. This finding is consistent with Refs. and. The Krook boundary layers damp the distributions back to being Maxwellian, generally reducing kinetic energy at the edges of the plasma. Kinetic energy is lost from the system primarily via this process. Electron trapping in IAWs serves to flatten the peak of the distribution in velocity space. Electrons then propagate to either boundary (note that generally v φ ± v tr,e v φ, and therefore resonant electrons have a larger spread in velocity than resonant ions and will reach the boundaries sooner) and are damped back to being Maxwellian. This damping of the perturbed distribution results in a value of K e below that of a quiescent plasma (recall that K e,i are spatially-averaged quantities), apparent in the negative value of K e shown in Fig. 6. The time taken for the reflectivity to recur, τ rec (introduced in Sec. III A ), is in agreement with the time taken in Fig. 6 for K j to grow, saturate, and return to zero (and therefore, presumably, for f j to return to a state close to the initial quiescent one); this is discussed further in Sec. IV. Examples of the ion and electron distributions typical of those during SBS are shown at snapshots in time (and for small samples in x) in Figs. 7 and 8, respectively; the velocity region has been restricted to that resonant with IAWs, and the distribution is shown as a deviation from the initial Maxwellian one. Figs. 7(a) and 8(a) show the particle distributions in a turbulence-free spatial region. The λ s -periodic nature of the SBS-driven IAW is evident in both species distributions. Figs. 7(b) and 8(b) show the distributions in a highly turbulent spatial region of

8 8 υ/υ ti υ φ υ/υ ti λ SBS υ φ +υ tr,i υ φ -υ tr,i (a) (b) x/λ De ~ f i FIG. 7. (Color online) Deviation of the ion distribution, f i, from Maxwellian, f M, where f i = [v ti/(z in i) ](f i f M ), shown in the velocity region resonant with IAWs, at ω pit = 8 using set S. (a) Spatial region where significant SBS is occurring [sampled region corresponds to grey box of Fig. (a)]. (b) Spatial region where turbulence is inhibiting SBS [sampled region corresponds to grey box of Fig. (c)]. Arrows indicate examples of vortices, defined in the text. v tr,i is calculated using measured eφ/t e =.65. υ φ υ/υ te υ/υ te λ SBS υ φ +υ tr,e υ φ -υ tr,e (b) (a) x/λ De ~ f e FIG. 8. (Color online) As Fig. 7, but here for the deviation of the electron distribution, f e = (v te/n e)(f e f M ). v tr,e is calculated using measured eφ/t e =.65. the IAW evolution, at which point the distributions show little evidence of a coherent plasma wave. The deviation of the distributions from Maxwellian are contained approximately within the regions bound by v φ ± v tr,j. Vortices in the ion distribution phase space are apparent in Fig. 7(b) (marked by arrows), suggestive of the formation of what have been referred to as kinetic electrostatic ion nonlinear (KEIN) waves, the ionic analog of KEEN waves 9. These structures, occurring at the onset of turbulence and coincident with the formation of largeamplitude solitons, remain small in amplitude and do not appear to impact directly the behavior of the reflectivity in our simulations. B. Set S: ZT e/t i = 5, smaller ν s, and larger δω s. Reflectivity and frequency detuning The full plasma parameters for the case discussed in this Section are given in Table I, listed under set S. The plasma reflectivity, R, and evolution of E x in k- space are shown in Fig. 9(a). This case is dominated by a feature not significant in the S case: the saturation of SBS by the dephasing of the local IAW from the ponderomotive force of the beating SBS light waves 7,8. This dephasing is due primarily to the trapping-induced nonlinear frequency shift δω, shown in 9(b), and is addressed in more detail later in this section. In S, v φ /v ti is sufficiently large such that ion kinetic effects are very weak, resulting in a positive value of δω dictated only by electron trapping. The magnitude of δω is larger in the perturbative limit for given φ s by a factor of or more than that of case S (see Table I), and is also of opposite sign. In Ref., it was found that the IAW subharmonic growth rate decreased sharply for α 5 in a D system. The growth rates γ l during the first period of strong subharmonic growth are shown in Fig., taken over the time window ω pi τ fit = [.4,.75] (see caption for details). In the case examined here, we find that near the first burst of reflectivity, decay rates are similar to the case S although decay appears to saturate before the system becomes strongly turbulent until a second subharmonic growth phase at ω pi t. Decay rates are higher than expected based on the results of Ref., where IAW decay was observed to be strongly suppressed at α = 5 until eφ s /T e.5. However, in the current work, the pump IAW mode is generally not exactly monochromatic and nonlinear mode frequencies evolve substantially, perhaps facilitating a more resonant driving of subharmonic modes than in Ref.. The growth rates in Fig. appear to lack clear symmetry about k l = k s (n /) over the intervals k l = [(n )k s, nk s ] present in Fig. 5, displaying a bias to lower k that is apparent in the mode amplitudes shown in Fig. 9. With similar plasma parameters, this feature was not observed for freely-propagating IAWs in a periodic system in Ref.. However, a similar effect was observed for instability occurring during SRS in Ref. 6 (i.e., asymmetric sideband growth rates about a carrier wave). We attribute this to difficulties in measuring the strictly linear growth phase of subharmonics in the presence of rapidly varying conditions.

9 9 k/k s ω/ω s L (%) E ~ x (k) (a) R, log. scale R, lin. scale (b) ω pi t Reflectivity, R Theory ω s NL ω max FIG. 9. (Color online) (a) Using set S, the reflectivity of the entire plasma versus time (right vertical axes, linear and logarithmic scale) and the changing composition of Fourier k modes in the longitudinal field of the IAW (left vertical axis and top color bar). SBS saturation is now due predominantly to the nonlinearity of the IAW frequency. (b) The deviation from the linear frequency of the resonant mode during SBS, ωs L, of i) ωs NL according to theory given in Table I, ii) ωs NL from simulation, and iii) the largest amplitude IAW mode, ω max, from simulation. We discuss now the role of the nonlinear frequency shift in saturating SBS in more detail. When the ponderomotive force of the light waves is in phase with the IAW, energy may be locally transferred efficiently from the pump to the scattered light and IAW. As the IAW amplitude grows, the phase of the IAW is shifted by δω = δω( φ ), and this may lead to a reduction in IAW amplitude, strongest when the phase mismatch between oscillator (IAW) and driver (ponderomotive force) is equal to π. Taking the (m j /)v moment of the Vlasov equation and neglecting the heat flow term Q = (/) j m j dv v f j / x, one obtains the local power transfer, P = j J j v,j B, (6) where J j is the charge current, v is the velocity in the longitudinal direction [recall f j = f j (t, x, v)], v,j is the transverse flow velocity (in the direction of the polarization of the electric field of the laser), and B is the magnetic field of the electromagnetic waves. When P >, energy is transferred to the IAW as either kinetic or potential energy. After averaging over fast-phase temporal oscillations, it is easy to show that P Φ (E E )/ x, γ l /ω s L Growth Decay Modes driven resonantly via SBS Vlasov simulation -wave fluid theory k l /k s FIG.. (Color online) Growth rates γ l for set S, measured using the fitting method employed in Fig. 5 for time ω piτ fit = [4, 75] sampled across x t=t /λ De = [5, ] where x = x v φ (t t ), ω pit = 4, and v φ =.c i. The -wave fluid theory is given by Eq. (6) using eφ/t e τfit =.. where E and E are the electric field amplitudes of the laser and scattered light, respectively. A similar diagnostic tool was applied successfully to SRS,. P is plotted for the entire simulated system for cases S and S in Figs. and, respectively. Averaging of P over λ s has been performed in order to suppress the sub-λ s changes of sign of P due to harmonic generation. In Fig., P is positive during the rise of the first peak in the reflectivity R. At the first saturation of R occurring at ω pi t, P is scrambled due to the growth of subharmonic modes, and there is no longer an effective driving of the IAW. The IAW amplitude then collapses, and R continues to fall even when P. However, in Fig., the oscillations in R are closely correlated with changes of sign in P, apparent even during the rapid oscillations occurring for ω pi t. We conclude that the behavior of R is dominated in case S by the spatiotemporal variations of δω.. The absence of significant particle heating Electrostatic, electromagnetic, kinetic, and sloshing energies are shown for the case S in Fig., plotted previously for the case S in Fig. 6 (the plotted quantities are defined in Sec. III A ). Despite U ES exceeding the value attained in the case S by a factor of, in the case S there is no evidence of significant particle heating: throughout the simulation, the kinetic energy of each species is dictated by the sloshing motion, i.e. K j U sj. Even when the plasma becomes more strongly turbulent (ω pi t ), significant heating occurs of neither electrons nor ions. There are at least two reasons why the heating is so weak in this case: i) The ponderomotive drive from the beating of the light waves is weaker in S than in S due to the reduced laser intensity. While this weaker force produces a value of U ES in S that exceeds that of S due to the difference in ν s, it may be

10 -5 Power transfer, P/neTeωpi ( ) x/λde R, lin. scale ωpi t Reflectivity, R R, log. scale U/neTe. -5 R, log. scale - x/λde ωpi t R, lin. scale.9.4 Reflectivity, R UEM Ki Usi ωpi t Power transfer, P/neTeωpi ( ) UES Ke Use FIG.. (Color online) As Fig. 6, but for the case S. Note the differing multiplicative factors applied to UES in Figs. 6 and. In this case, there is little heating and the changes in kinetic energies are dominated by sloshing.. FIG.. (Color online) For set S, the evolution of the power transfer, P (left vertical axis and top color bar) defined by Eq. (6) with the plasma reflectivity, R (right vertical axis, linear and logarithmic scale), superimposed. Comparison of P and R indicates saturation of R is not caused predominantly in this case by dephasing of the IAW and ponderomotive drive of the light waves FIG.. (Color online) As Fig., but for the case S. Comparison of P and R indicates saturation of R is caused predominantly in this case by dephasing of the IAW and ponderomotive drive of the light waves, and produces rapid oscillations in R late in time. that this difference in linear damping has little impact on the nonlinear phase of the simulations. As a consequence, the IAWs would be simply more weakly driven, leading to less heating. ii) There are fewer ions in the resonant region in S than in S. However, this second possibility has no direct effect on electron heating, which is significant in S but not S. IV. PRIOR WORK AND DISCUSSION Previously, simulation-based efforts to understand the nonlinear saturation mechanisms of SBS have been performed using D and D PIC simulations. In the work of Cohen et al.8,9, a fluid electron model (Boltzmann) was adopted and coupled to a kinetic ion PIC description. Such an approach allows greatly reduced computational effort, but by design does not describe electron kinetic effects such as electron Landau damping and the contribution of the electrons to the nonlinear frequency shift. Because of the differences in simulation methodology, timescales, and parameters (in particular, the D results of Cohen et al. were in a regime of significant pump depletion), quantitative comparisons with the work presented here are difficult. However, some qualitative similarities are clear, such as the presence of significant IAW decay and a correlation of the onset of significant decay with a saturation in SBS. Riconda et al.6 used a similar numerical approach to Cohen et al. to study SBS and also observed IAW decay. It is no surprise that we find electron kinetic effects are important in a collisionless plasma at high α. However, the results of Refs.,,, 4, and the work presented here suggest strongly that electron kinetic effects are important in determining the strength of the IAW mode-mode coupling and therefore harmonic generation and subharmonic decay even at low α. In Divol et al.7 (in which the same code as in the works of Cohen et al. was employed), a distinction was drawn between a frequency shift arising due to trapping in a quasimonochromatic wave and a frequency shift arising due to a quasi-linear modification of the local ion distribution (which may have previously been caused by trapping) was drawn. In Ref. 7, IAW decay to longer wave lengths occurred and was determined to have an impact upon

11 reflectivity that while substantial was generally weaker than the effect of detuning due to the nonlinearity (and resulting spatial inhomogeneity) of the IAW frequency. The absence of electron kinetic effects in Ref. 7 that at lower ZT e /T i may reduce the net frequency shift (in addition to enhancing IAW decay) perhaps played a roll in the relative strengths of IAW decay and nonlinear dephasing. By performing D simulations, we have explicitly neglected IAW decay into modes with a non-zero transverse wave number component. From fluid theory of IAW decay 9,, it is expected that such decay channels will be faster than modes parallel to the carrier waves, perhaps enhancing IAW decay as a saturation mechanism of SBS compared to D systems. However, kinetic effects such as the anisotropic flattening of the species distributions and resulting anisotropic damping may modify such a picture significantly. D and D PIC simulations were compared by Cohen et al. 5,8,9. The IAW amplitude following saturation in the SBS reflectivity was found to be lower in D than in D, with IAW decay indeed occurring fastest for non-parallel modes. As a general remark, we observe that the SBS reflectivity in sapristi appears significantly lower than the reflectivity in PIC simulations published elsewhere. We speculate that this is due to the kinetic electron treatment employed here (which introduces stronger nonlinearity than a fluid electron model) and the noise-induced field fluctuations in PIC simulations that mean less growth is needed in order to reach IAW amplitudes that cause significant SBS. It is apparent from this work and others that there are multiple coexisting and effective mechanisms of saturation of SBS. Which mechanism is dominant will depend upon the laser intensity, but likely also upon the plasma parameters; further work, perhaps using a D Vlasov-Maxwell code, would clarify this point. V. CONCLUSIONS Using fully-kinetic simulations, we have shown that IAW decay occurs in systems where IAWs are excited by a highly time-varying ponderomotive drive arising from SBS. From these simulations, we have been able to extract for the first time a growth rate of the decay modes that is in agreement with that of freely-propagating IAWs, allowing the unambiguous identification of the decay process. This decay can act as an effective saturation mechanism for SBS in D systems, resulting in a crash in IAW amplitude that provokes a loss of plasma reflectivity. The decay occurs more readily in D for lower ZT e /T i. At ZT e /T i = 5, dephasing of the driven IAW from the ponderomotive force of the laser and SBS light wave is the dominant saturation mechanism. ACKNOWLEDGMENTS We gratefully acknowledge fruitful discussions with Bruce I. Cohen on a range of topics relevant to this work. We are thankful to Bill Arrighi for computer science support essential to the completion of this work. This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under Contract DE-AC5-7NA744 and funded by the Laboratory Research and Development Program at LLNL under project tracking codes -ERD-6 and 5-ERD-8. Benjamin J. Winjum acknowledges support also from the U.S. DOE under Grant Nos. DE- NA8 and DE-FC-4ER APPENDIX: SIMULATION SETUP We describe here the setup of the numerical simulations presented in this work, summarized in Fig.. The interaction of laser light with the plasma is simulated using the kinetic code sapristi, which solves here the collisionless D Vlasov-Maxwell system of equations. Distribution functions for each plasma species (in this case, electrons and one ion species with a physically correct mass ratio) are evolved using a semi-lagrangian scheme with a time step t =.ωpe, sufficiently small to resolve electron kinetic (wave-particle) effects, with 7 sub-cycled steps for the electromagnetic portion of the calculation. An in-depth discussion of the code is given in Refs. 5 and 4. In order to resolve v tr,j, velocity meshes of 4 and 48 points for the electrons and ions, respectively, are chosen, spaced evenly across the ranges [ 8v tj, 8v tj ] with neutral boundary conditions. This is sufficient to resolve the complex kinetic phenomena occurring in our simulations across the relevant range of φ. A spatial resolution of 64 grid points per λ s corresponding to x.6λ De was chosen, adequate to describe accurately the nonlinear IAW dynamics over the simulated time duration. The total number of spatial grid points across L was.8 4. The basic simulation geometry is shown in Fig.. The plasma fluctuations at the edge of the simulated system are damped using a Krook operator in the Vlasov equation, ramped up smoothly from an effective damping rate of to a maximum of ω pe across grid points at either edge of the plasma. This Krook operator is chosen to be conservative of particle number but not energy; the species distribution functions are damped back to their initial Maxwellian states, and the boundary layers may be viewed as a thermal bath. Laser light (assumed linearly polarized) of intensity I x= and frequency ω = π/λ traveling in the direction of increasing x is emitted via an antenna composed of a pair of current sheets near the x = boundary. The phasing of the currents ensures that the antenna emits in one direction only. Because the Vlasov-Maxwell solution method employed in sapristi is noiseless to machine pre-